The standard administrative subdivision of Sweden is Country -> County -> Municipality. All of these have legal significance and their borders have historically been determined by political decisions. Alongside the standard administrative subdivisions Statistics Sweden (Statistiska Centralbyrån) as well as other government agencies have created subdivisions for statistical purposes, such as the Regional Statistical Areas (RegSO), Demographic Statistical Areas (DeSO) and local labour markets (lokala arbetsmarknader). I will in this post take a special interest in the local labour market subdivision. The material is quite technical (admittedly more than initially intended) so I envision the primary target audience to be the people responsible for developing and maintaining the local labour market standard in various countries.

The local labour markets are created by first determining which municipalities can count as “local centers”. A local center is a municipality in which

After the local centers are created all remaining municipalities are assigned to the labour market created by the local center that most of its commuting residents commute to. (Construction and use of labour market areas in Sweden, Statistics Sweden 2010).

Quoting directly from the aforementioned SCB report: “The purpose of local labour markets is to describe the functioning of the labour market for geographical areas that are relatively independent of the outside world with respect to supply and demand of labour” (translation my own, original source in Swedish).

This post will ask and answer the question: Does this method of subdivision work by testing whether or not the local labour markets are mostly independent of the outside world. To perform this test this I will analyse how this subdivision type interacted with the COVID-19 pandemic.

The COVID-19 pandemic - A novel stress test

In 2020 the COVID pandemic swept over the world, with Sweden being no exception. It is perhaps the best monitored and documented pandemic in the history of the world. In particular, the granular data collected by the Swedish Public Health Agency (Folkhälsomyndigheten) provides an excellent opportunity to test if the labour market regions live up to their purpose of being “relatively independent of the outside world”

If labour market regions are truly independent economic units we would expected COVID-19 spread patterns between adjacent labour markets to be less correlated than say, between counties (whose borders are not drawn with the explicit purpouse of defining self contained economic zones)

This formulation will lead the layman to ask questions such as “What counts as ‘less correlated’” and the expert asking questions such as “How do we define ‘correlation’”. I will make explicit my modeling assumptions later in the post, but first, let us visualise the spread of COVID through Swedish municipalities. While I wish to keep the amount of code displayed to a minimum, credit is due to Filip Wästberg for the creation of the R package “swemaps2”, which I make heavy use of.

library(remotes)
remotes::install_github("filipwastberg/swemaps2")

Now that we have installed swemaps2 we can make a map of Swedens labour market regions. They changed ever so slightly during the pandemic, so we will have to account for that in our modeling.

While certainly pretty it would be hard to discern any meaningful patterns by just peering over these visualisations, even if we were to overlay the borders of our subdivisions of interest. This creates the need for a good statistical model to estimate the effect adjacent labour markets have on each other. We also need to answer the question of what counts as a strong relationship, and how this relationship will be measured.

Modeling

The most appropriate model from my limited experience and understanding would probably be zero-inflated negative binomial regression. However, to avoid having to verify complicated modeling assumptions in an area where I lack expertise, as well as the computational cost of fitting such a model I have instead resorted to the following:

We first define our modeling target, which will be the log geometric change in cases in a municipality between time (measured in weeks) \(t\) and \(t-1\). We call this \([\text{log case ratio}]_{i,t} = \log\frac{\text{cases in municipality i at time t}}{\text{cases in municipality i at time t-1}}\). The reason we set our modeling target to the log geometric change in cases is partially to avoid having to model count data and partially to avoid (or at least reduce) the autocorrelation of case counts over time. With our modeling target defined we can now define our model, which will be

\([\text{log case ratio}]_{i,t} = \alpha + \beta\mathbf{1}_{time = t} + \gamma\mathbf{1}_{\text{labour market of municipality i}} + \epsilon_{i,t}\).

This represents a fixed effects model with a time effect and a labour market effect.

We have defined our modeling target, great. But now to the question of what counts as a “weak relationship” between adjacent labour markets. A priori we have very limited information about how such a relationship would look, so to gain information we will run the same model but using adjacent counties instead. In theory county borders should be largely unrelated (or at least less related) to labour market, since most county borders stem from the Axel Oxenstierna county reform of 1614. While Oxenstierna undeniably was a great statesman I doubt that he managed to predict the structure of the Swedish labour market in the 21st century, so his borders should serve as a good benchmark. This gives us the “Oxenstierna” or county model

\([\text{log case ratio}]_{i,t} = \alpha + \beta\mathbf{1}_{time = t} + \gamma\mathbf{1}_{\text{county of municipality i}} + \epsilon_{i,t}\).

We will not pay too much attention to the standard modelling assumptions of linear regression (conditionally independent response variable, iid normal errors with constant variance) since we will not be performing frequentist tests or relying on the likelihood. Instead, we will measure the correlation of the models residuals to the log case count in adjacent labour markets/counties.

It may seem counterintuitive, but if labor market regions are truly functioning as independent units, we would expect adjacent regions to have less predictive power for COVID-19 spread compared to the predictive power of a county based model.

To make this comparison we will need to handle both the fact that the amount of labour market regions changed and that the constituent municipalities changed during the pandemic. There were 69 labour market regions in 2020, which shrunk to 67 for 2021 - 2023 (perhaps due to decreased travel caused by the pandemic). We also have to handle that 2020 was a leap year with 53 weeks. I have also decided to use the 2023 labour market regions for 2024 since the updated ones were not available at the time of writing. Finally, we will also have to remove Gotland since it both as a labour market and as a county lacks adjacent labour markets / counties, making models including it un-identifiable.

Several hundred lines of cumbersome data wrangling later and we are finally ready to run our models, which are shown in the code below

We will start by computing Spearmans \(\rho\) for each municipality and plotting it

source("correlation_functions.R")

la_anova_model <- lm(log(case_ratios) ~ as.factor(time) + as.factor(la_region), data = compound_la_adjacency_frame)
county_anova_model <- lm(log(case_ratios) ~ as.factor(time) + as.factor(county), data = compound_county_adjacency_frame)

compound_la_adjacency_frame$fe_residuals <- resid(la_anova_model)
compound_county_adjacency_frame$fe_residuals <- resid(county_anova_model)


la_municipal_correlations <- municipal_correlations(compound_la_adjacency_frame)
county_municipal_correlations <- municipal_correlations(compound_county_adjacency_frame)

la_municipal_correlations <- merge(la_municipal_correlations, municipality, by.x = "municipal_code", by.y = "kn_kod")
county_municipal_correlations <- merge(county_municipal_correlations, municipality, by.x = "municipal_code", by.y = "kn_kod")

la_correlation_plot <- ggplot(la_municipal_correlations, 
    aes(geometry=geometry, fill=srho)) + 
    geom_sf() + 
    common_fill_scale +
    theme_swemap2() + 
    theme(
        plot.title = element_text(size = 10, hjust = 0.5),
        plot.margin = margin(5, 5, 5, 5),
        aspect.ratio = 2  # For Sweden's proportions
    ) +
    labs(title = "LA model")

county_correlation_plot <- ggplot(county_municipal_correlations, 
    aes(geometry=geometry, fill=srho)) + 
    geom_sf() + 
    common_fill_scale +
    theme_swemap2() + 
    theme(
        plot.title = element_text(size = 10, hjust = 0.5),
        plot.margin = margin(5, 5, 5, 5),
        aspect.ratio = 2
    ) +
    labs(title = "Oxenstierna/county model")

combined_plot <- la_correlation_plot + county_correlation_plot + 
    plot_layout(guides = "collect", widths = c(1, 1)) & 
    theme(
        plot.title = element_text(size = 12, hjust = 0.5),
        legend.position = "right",  # Move legend to right
        legend.justification = "left"  # Align legend with plots
    )
combined_plot <- combined_plot + 
    plot_annotation(
        title = "Spearman's correlation coefficients for municipalities",
        theme = theme(
            plot.title = element_text(size = 14, hjust = 0.5, margin = margin(b = 20))
        )
    )

combined_plot

Again while the map form of visualisation certainly is pleasing to the eye a histogram can’t hurt.

Conclusions

The rank correlations for the county based model show a fatter right tail and mean shifted to the right of the labour market model, consistent with that labour markets should be mostly independent entities. Directionally, the relationship is consistent with SCBs goals, and a paired T-test of the rank correlations gives us a p value in the \(10^{-12} - 10^{-11}\) range indicating it is unlikely due to randomness.

t.test(la_municipal_correlations$srho, county_municipal_correlations$srho, paired = TRUE)

It is however hard to interpret if the effect size is any meaningful. Initially I tried adding the adjacent cases as a parameter to our linear model, but getting the dataset to adhere in any way to the necessary modelling assumptions was almost impossible. Given more time I would have created the a Zero Inflated Negative Binomial model.

Does the Labour Market Subdivision Work?

Introduction

The COVID-19 pandemic - A novel stress test

Modeling

Conclusions